Let $\,X:=\big( [ \, 0,+\infty\, [ \, \, \times \, \,[ \, 0 \,, 1\, ]\big)\, \, \cup \, \, \big( ] \, -\infty,0\, ] \, \, \times \, \,\{ \, 0 \,\}\big) \,$ as a topological subspace of $\mathbb{R}^2$ and let $p_2 :X \, \rightarrow \mathbb{R} $ be the projection on the second coordinate, Prove that:
1) $p_2$ is an identification (or quotient map).
2) $p_2$ is not open.
3) $p_2$ is closed.
$\mathbf{My \, \, attempt: }$
For 2) i used the set : $\quad ]\,-1, 1 \, [ \, \, \times \, \, ]\, -1, \, 2 \,[ \,$ that is open in $\mathbb{R}^2\,$ and intersects $X$ in $]\,-1, 0\,] \, \times \, \{0\} \, \, \cup \, \, [\, 0, 1\,[ \, \times\, [\, 0,\,1] \, $ that has a closed image trough $p_2$. Now my idea was to prove 3) and then use that $p_2$ is closed, continue and surjective to prove 1). To prove 3) i'm using the fact that any closed set in $X$ is closed in $\mathbb{R}^2$, has a bounded coordinate along the y-axis, therefore the image trough $p_2$ is compact and so closed in $\mathbb{R}$ (i'm sorry for not being totally clear on this point but it's not in the main scope of why i'm asking this question).
$\mathbf{my \, \,Question: }$ Why is $p_2 $ surjective?